Dr. Hirst on Equally Attracting Bodies. 177 



other, and if (p'l) be a third surface inverse to either of the former, 

 the pole will be attracted in the same manner by the two surfaces 

 r= \^ ppi and r,= \^pp\, whose radii vectores are, respectively, 

 mean proportionals between p and pi, and between p and p\. 

 For by hypothesis, 



2u^ = v — ^1 ; 

 whence, diflferentiating, squaring, &c., due regard being had to 

 the equation (15), which by hypothesis is here satisfied, we easily 

 find that 



Xdd) sin2 6\d<f>) \dd) ^ sin^ e\d(f>) ' 

 since each is equal to 



^ Lw^y ^ sin2^w</>; ^\dd) sm^e\d(f>J J ' 



or, as otherwise expressed, 



tan^-\/r= tan'^'«|r| = ^(tan^OT + tan^Wi), 

 where ^/r, ■^^, ct, ■st, represent respectively the acute angles be- 

 tween any radius vector and the corresponding normals to the 

 surfaces (r), [r^), (p), (pj). 



22. From the above it appears that there is no immediate 

 relation between the attractions of either of the surfaces (r), (r,) 

 and those of (p) and (pi) ; nevertheless, from the theorems of the 

 last two articles, we may deduce the coi'ollary, that when the 

 surfaces (r) and (r,) not only attract equally, but also have their 

 corresponding normal vector-planes perpendicular to each other, 

 the surfaces (p) and (p,) enjoy the same properties, and then 



tan^ i|r = tan^ ilr, = i tan^ ■o" = ^^ tan^ ■ctj. 

 For instance, since, from art. 16, the two surfaces 



p = 7tan'»^, 



attract the pole equally, and at the same time have their corre- 

 sponding normal vector-planes perpendicular to each other, it 

 follows from the above that the surfaces 



r = c.e^ .tan'^^, 



r, = e, . e ^ . tan^ jr 



enjoy the same properties, and in this case 



tan2-«!r=tan*i|r,= • tan2'cr = i tan2CT, = — - • -^-^a. 



[To be continued.] 

 Phil. Mag. S. 4. Vol. 16. No. 106. Sept. 1858. N 



